
Next, for \(b_n:=\sqrt{\beta _n}\cdot n^\alpha \le a_n\)  and by [1]},
\(\Vert f_n-f_n^{\prime }\Vert _{L^p}&\le \Big \Vert \int _{-\infty }^{\infty } \big (f_n( \raisebox {-0.5ex}{\scalebox {2}{\cdot }})-f_n( \raisebox {-0.5ex}{\scalebox {2}{\cdot }}-y)\big ) \phi _{\beta _n}(y) \mathbb {1}_{\lbrace |y|\le b_n\rbrace } \,\mathrm {d}y\Big \Vert _{L^p}\\&\quad +\Big \Vert \int _{-\infty }^{\infty } \big (f_n( \raisebox {-0.5ex}{\scalebox {2}{\cdot }})-f_n( \raisebox {-0.5ex}{\scalebox {2}{\cdot }}-y)\big ) \phi _{\beta _n}(y) \mathbb {1}_{\lbrace |y|> b_n\rbrace } \,\mathrm {d}y\Big \Vert _{L^p}\\&\le \big \Vert \phi _{\beta _n}( \raisebox {-0.5ex}{\scalebox {2}{\cdot }})\mathbb {1}_{\lbrace | \raisebox {-0.5ex}{\scalebox {2}{\cdot }}|\le b_n\rbrace }\big \Vert _{L^1} \cdot \sup _{|h|\le b_n} \big \Vert f_n( \raisebox {-0.5ex}{\scalebox {2}{\cdot }})-f_n( \raisebox {-0.5ex}{\scalebox {2}{\cdot }}+h)\big \Vert _{L^p}\\&\quad + \big \Vert \phi _{\beta _n}( \raisebox {-0.5ex}{\scalebox {2}{\cdot }})\mathbb {1}_{\lbrace | \raisebox {-0.5ex}{\scalebox {2}{\cdot }}|> b_n\rbrace }\big \Vert _{L^1} \cdot \sup _{|h|> b_n} \big \Vert f_n( \raisebox {-0.5ex}{\scalebox {2}{\cdot }})-f_n( \raisebox {-0.5ex}{\scalebox {2}{\cdot }}+h)\big \Vert _{L^p}.\) 
